A fast parallel algorithm for finding Hamiltonian cycles in dense graphs

Suppose than 0 < eta < 1 is given. We call a graph, G, on n vertices an eta-Chvatal graph if its degree sequences d(1) <= d(2) <= ... <= d(n) satisfies: for k < n/2, d(k) <= min {k + eta n. n/2} implies d(n-k-eta n) >= n - k. (Thus for eta = 0 we get the well-known Chvata graphs.) An NC(4)-algorithm is presented which accepts as input an eta-Chvatal graph and produces a Hamiltonian cycle in G as an output. This is a significant improvement on the previous best NC-algorithm for the problem, which finds a Hamiltonian cycle only in Dirac graphs (delta(G) >= n/2 where delta(G) is the minimum degree in G). (C) 2008 Elsevier B.V. All rights reserved.